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Introduction to Information Retrieval
Introduction to
Information Retrieval
Hinrich Schütze and Christina Lioma
Lecture 6: Scoring, Term Weighting, The Vector
Space Model
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Introduction to Information Retrieval
Overview
❶
Recap
❷
Why ranked retrieval?
❸
Term frequency
❹
tf-idf weighting
❺
The vector space model
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Introduction to Information Retrieval
Outline
❶
Recap
❷
Why ranked retrieval?
❸
Term frequency
❹
tf-idf weighting
❺
The vector space model
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Introduction to Information Retrieval
Heaps’ law
Vocabulary size M as a
function of collection size
T (number of tokens) for
Reuters-RCV1. For these
data, the dashed line
log10M =
0.49 ∗ log10 T + 1.64 is the
best least squares fit.
Thus, M = 101.64T0.49
and k = 101.64 ≈ 44 and
b = 0.49.
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Introduction to Information Retrieval
Zipf’s law
The most frequent term
(the) occurs cf1 times, the
second most frequent term
(of) occurs
times, the third most
frequent term (and) occurs
times etc.
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Introduction to Information Retrieval
Dictionary as a string
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Introduction to Information Retrieval
Gap encoding
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Introduction to Information Retrieval
Variable byte (VB) code
 Dedicate 1 bit (high bit) to be a continuation bit c.
 If the gap G fits within 7 bits, binary-encode it in the 7
available bits and set c = 1.
 Else: set c = 0, encode high-order 7 bits and then use one or
more additional bytes to encode the lower order bits using
the same algorithm.
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Introduction to Information Retrieval
Gamma codes for gap encoding





Represent a gap G as a pair of length and offset.
Offset is the gap in binary, with the leading bit chopped off.
Length is the length of offset.
Encode length in unary code
The Gamma code is the concatenation of length and offset.
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Introduction to Information Retrieval
Compression of Reuters
data structure
dictionary, fixed-width
dictionary, term pointers into string
∼, with blocking, k = 4
∼, with blocking & front coding
collection (text, xml markup etc)
collection (text)
T/D incidence matrix
postings, uncompressed (32-bit words)
postings, uncompressed (20 bits)
postings, variable byte encoded
postings, γ encoded
size in MB
11.2
7.6
7.1
5.9
3600.0
960.0
40,000.0
400.0
250.0
116.0
101.0
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Introduction to Information Retrieval
Take-away today
 Ranking search results: why it is important (as opposed to just
presenting a set of unordered Boolean results)
 Term frequency: This is a key ingredient for ranking.
 Tf-idf ranking: best known traditional ranking scheme
 Vector space model: One of the most important formal
models for information retrieval (along with Boolean and
probabilistic models)
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Introduction to Information Retrieval
Outline
❶
Recap
❷
Why ranked retrieval?
❸
Term frequency
❹
tf-idf weighting
❺
The vector space model
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Introduction to Information Retrieval
Ranked retrieval
 Thus far, our queries have all been Boolean.
 Documents either match or don’t.
 Good for expert users with precise understanding of their
needs and of the collection.
 Also good for applications: Applications can easily consum
1000s of results.
 Not good for the majority of users
 Most users are not capable of writing Boolean queries . . .
 . . . or they are, but they think it’s too much work.
 Most users don’t want to wade through 1000s of results.
 This is particularly true of web search.
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Introduction to Information Retrieval
Problem with Boolean search: Feast or famine
 Boolean queries often result in either too few (=0) or too
many (1000s) results.
 Query 1 (boolean conjunction): [standard user dlink 650]
 → 200,000 hits – feast
 Query 2 (boolean conjunction): [standard user dlink 650 no
card found]
 → 0 hits – famine
 In Boolean retrieval, it takes a lot of skill to come up with a
query that produces a manageable number of hits.
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Introduction to Information Retrieval
Feast or famine: No problem in ranked retrieval




With ranking, large result sets are not an issue.
Just show the top 10 results
Doesn’t overwhelm the user
Premise: the ranking algorithm works: More relevant results
are ranked higher than less relevant results.
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Introduction to Information Retrieval
Scoring as the basis of ranked retrieval
 We wish to rank documents that are more relevant higher
than documents that are less relevant.
 How can we accomplish such a ranking of the documents in
the collection with respect to a query?
 Assign a score to each query-document pair, say in [0, 1].
 This score measures how well document and query “match”.
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Introduction to Information Retrieval
Query-document matching scores
 How do we compute the score of a query-document pair?
 Let’s start with a one-term query.
 If the query term does not occur in the document: score
should be 0.
 The more frequent the query term in the document, the
higher the score
 We will look at a number of alternatives for doing this.
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Introduction to Information Retrieval
Take 1: Jaccard coefficient
 A commonly used measure of overlap of two sets
 Let A and B be two sets
 Jaccard coefficient:
 JACCARD (A, A) = 1
 JACCARD (A, B) = 0 if A ∩ B = 0
 A and B don’t have to be the same size.
 Always assigns a number between 0 and 1.
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Introduction to Information Retrieval
Jaccard coefficient: Example
 What is the query-document match score that the Jaccard
coefficient computes for:
 Query: “ides of March”
 Document “Caesar died in March”
 JACCARD(q, d) = 1/6
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Introduction to Information Retrieval
What’s wrong with Jaccard?
 It doesn’t consider term frequency (how many occurrences a
term has).
 Rare terms are more informative than frequent terms.
Jaccard does not consider this information.
 We need a more sophisticated way of normalizing for the
length of a document.
 Later in this lecture, we’ll use
(cosine) . . .
 . . . instead of |A ∩ B|/|A ∪ B| (Jaccard) for length
normalization.
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Introduction to Information Retrieval
Outline
❶
Recap
❷
Why ranked retrieval?
❸
Term frequency
❹
tf-idf weighting
❺
The vector space model
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Introduction to Information Retrieval
Binary incidence matrix
Anthony Julius
and
Caesar
Cleopatra
ANTHONY
BRUTUS
CAESAR
CALPURNIA
CLEOPATRA
MERCY
WORSER
...
1
1
1
0
1
1
1
The
Hamlet
Tempest
1
1
1
1
0
0
0
0
0
0
0
0
1
1
Othello
0
1
1
0
0
1
1
Macbeth
...
0
0
1
0
0
1
1
1
0
1
0
0
1
0
Each document is represented as a binary vector ∈ {0, 1}|V|.
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Introduction to Information Retrieval
Binary incidence matrix
Anthony Julius
and
Caesar
Cleopatra
ANTHONY
BRUTUS
CAESAR
CALPURNIA
CLEOPATRA
MERCY
WORSER
...
157
4
232
0
57
2
2
73
157
227
10
0
0
0
The
Hamlet
Tempest
0
0
0
0
0
3
1
Othello
0
2
2
0
0
8
1
Macbeth
...
0
0
1
0
0
5
1
1
0
0
0
0
8
5
Each document is now represented as a count vector ∈ N|V|.
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Introduction to Information Retrieval
Bag of words model
 We do not consider the order of words in a document.
 John is quicker than Mary and Mary is quicker than John
are represented the same way.
 This is called a bag of words model.
 In a sense, this is a step back: The positional index was able
to distinguish these two documents.
 We will look at “recovering” positional information later in
this course.
 For now: bag of words model
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Introduction to Information Retrieval
Term frequency tf
 The term frequency tft,d of term t in document d is defined
as the number of times that t occurs in d.
 We want to use tf when computing query-document match
scores.
 But how?
 Raw term frequency is not what we want because:
 A document with tf = 10 occurrences of the term is more
relevant than a document with tf = 1 occurrence of the
term.
 But not 10 times more relevant.
 Relevance does not increase proportionally with term
frequency.
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Introduction to Information Retrieval
Instead of raw frequency: Log frequency
weighting
 The log frequency weight of term t in d is defined as follows
 tft,d → wt,d :
0 → 0, 1 → 1, 2 → 1.3, 10 → 2, 1000 → 4, etc.
 Score for a document-query pair: sum over terms t in both q
and d:
tf-matching-score(q, d) =  t∈q∩d (1 + log tft,d )
 The score is 0 if none of the query terms is present in the
document.
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Introduction to Information Retrieval
Exercise
 Compute the Jaccard matching score and the tf matching
score for the following query-document pairs.
 q: [information on cars] d: “all you’ve ever wanted to know
about cars”
 q: [information on cars] d: “information on trucks,
information on planes, information on trains”
 q: [red cars and red trucks] d: “cops stop red cars more
often”
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Introduction to Information Retrieval
Outline
❶
Recap
❷
Why ranked retrieval?
❸
Term frequency
❹
tf-idf weighting
❺
The vector space model
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Introduction to Information Retrieval
Frequency in document vs. frequency in
collection
 In addition, to term frequency (the frequency of the term
in the document) . . .
 . . .we also want to use the frequency of the term in the
collection for weighting and ranking.
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Introduction to Information Retrieval
Desired weight for rare terms
 Rare terms are more informative than frequent terms.
 Consider a term in the query that is rare in the collection
(e.g., ARACHNOCENTRIC).
 A document containing this term is very likely to be
relevant.
 → We want high weights for rare terms like
ARACHNOCENTRIC.
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Introduction to Information Retrieval
Desired weight for frequent terms
 Frequent terms are less informative than rare terms.
 Consider a term in the query that is frequent in the
collection (e.g., GOOD, INCREASE, LINE).
 A document containing this term is more likely to be
relevant than a document that doesn’t . . .
 . . . but words like GOOD, INCREASE and LINE are not sure
indicators of relevance.
 → For frequent terms like GOOD, INCREASE and LINE, we
want positive weights . . .
 . . . but lower weights than for rare terms.
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Introduction to Information Retrieval
Document frequency
 We want high weights for rare terms like ARACHNOCENTRIC.
 We want low (positive) weights for frequent words like
GOOD, INCREASE and LINE.
 We will use document frequency to factor this into
computing the matching score.
 The document frequency is the number of documents in
the collection that the term occurs in.
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Introduction to Information Retrieval
idf weight
 dft is the document frequency, the number of documents
that t occurs in.
 dft is an inverse measure of the informativeness of term t.
 We define the idf weight of term t as follows:
(N is the number of documents in the collection.)
 idft is a measure of the informativeness of the term.
 [log N/dft ] instead of [N/dft ] to “dampen” the effect of idf
 Note that we use the log transformation for both term
frequency and document frequency.
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Introduction to Information Retrieval
Examples for idf
 Compute idft using the formula:
term
calpurnia
animal
sunday
fly
under
the
dft
1
100
1000
10,000
100,000
1,000,000
idft
6
4
3
2
1
0
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Introduction to Information Retrieval
Effect of idf on ranking
 idf affects the ranking of documents for queries with at
least two terms.
 For example, in the query “arachnocentric line”, idf
weighting increases the relative weight of ARACHNOCENTRIC
and decreases the relative weight of LINE.
 idf has little effect on ranking for one-term queries.
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Introduction to Information Retrieval
Collection frequency vs. Document frequency
word
INSURANCE
TRY
collection frequency
document frequency
10440
10422
3997
8760
 Collection frequency of t: number of tokens of t in the
collection
 Document frequency of t: number of documents t occurs in
 Why these numbers?
 Which word is a better search term (and should get a
higher weight)?
 This example suggests that df (and idf) is better for
weighting than cf (and “icf”).
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Introduction to Information Retrieval
tf-idf weighting
 The tf-idf weight of a term is the product of its tf weight
and its idf weight.





tf-weight
idf-weight
Best known weighting scheme in information retrieval
Note: the “-” in tf-idf is a hyphen, not a minus sign!
Alternative names: tf.idf, tf x idf
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Introduction to Information Retrieval
Summary: tf-idf
 Assign a tf-idf weight for each term t in each document d:
 The tf-idf weight . . .
 . . . increases with the number of occurrences within a
document. (term frequency)
 . . . increases with the rarity of the term in the collection.
(inverse document frequency)
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Introduction to Information Retrieval
Exercise: Term, collection and document
frequency
Quantity
term frequency
Symbol Definition
tft,d
number of occurrences of t in
d
document frequency
dft
number of documents in the
collection that t occurs in
collection frequency
cft
total number of occurrences of
t in the collection
 Relationship between df and cf?
 Relationship between tf and cf?
 Relationship between tf and df?
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Introduction to Information Retrieval
Outline
❶
Recap
❷
Why ranked retrieval?
❸
Term frequency
❹
tf-idf weighting
❺
The vector space model
40
Introduction to Information Retrieval
Binary incidence matrix
Anthony Julius
and
Caesar
Cleopatra
ANTHONY
BRUTUS
CAESAR
CALPURNIA
CLEOPATRA
MERCY
WORSER
...
1
1
1
0
1
1
1
The
Hamlet
Tempest
1
1
1
1
0
0
0
0
0
0
0
0
1
1
Othello
0
1
1
0
0
1
1
Macbeth
...
0
0
1
0
0
1
1
1
0
1
0
0
1
0
Each document is represented as a binary vector ∈ {0, 1}|V|.
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Introduction to Information Retrieval
Count matrix
Anthony Julius
and
Caesar
Cleopatra
ANTHONY
BRUTUS
CAESAR
CALPURNIA
CLEOPATRA
MERCY
WORSER
...
157
4
232
0
57
2
2
73
157
227
10
0
0
0
The
Hamlet
Tempest
0
0
0
0
0
3
1
Othello
0
2
2
0
0
8
1
Macbeth
...
0
0
1
0
0
5
1
1
0
0
0
0
8
5
Each document is now represented as a count vector ∈ N|V|.
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Introduction to Information Retrieval
Binary → count → weight matrix
Anthony Julius
and
Caesar
Cleopatra
ANTHONY
BRUTUS
CAESAR
CALPURNIA
CLEOPATRA
MERCY
WORSER
...
5.25
1.21
8.59
0.0
2.85
1.51
1.37
3.18
6.10
2.54
1.54
0.0
0.0
0.0
The
Hamlet
Tempest
0.0
0.0
0.0
0.0
0.0
1.90
0.11
0.0
1.0
1.51
0.0
0.0
0.12
4.15
Othello
0.0
0.0
0.25
0.0
0.0
5.25
0.25
Macbeth
...
0.35
0.0
0.0
0.0
0.0
0.88
1.95
Each document is now represented as a real-valued vector of tf
idf weights ∈ R|V|.
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Introduction to Information Retrieval
Documents as vectors
 Each document is now represented as a real-valued vector
of tf-idf weights ∈ R|V|.
 So we have a |V|-dimensional real-valued vector space.
 Terms are axes of the space.
 Documents are points or vectors in this space.
 Very high-dimensional: tens of millions of dimensions when
you apply this to web search engines
 Each vector is very sparse - most entries are zero.
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Introduction to Information Retrieval
Queries as vectors
 Key idea 1: do the same for queries: represent them as
vectors in the high-dimensional space
 Key idea 2: Rank documents according to their proximity to
the query
 proximity = similarity
 proximity ≈ negative distance
 Recall: We’re doing this because we want to get away from
the you’re-either-in-or-out, feast-or-famine Boolean
model.
 Instead: rank relevant documents higher than nonrelevant
documents
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Introduction to Information Retrieval
How do we formalize vector space similarity?





First cut: (negative) distance between two points
( = distance between the end points of the two vectors)
Euclidean distance?
Euclidean distance is a bad idea . . .
. . . because Euclidean distance is large for vectors of
different lengths.
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Introduction to Information Retrieval
Why distance is a bad idea
The Euclidean distance of and
is large although the distribution
of terms in the query q
and the distribution of terms in the document d2 are very similar.
Questions about basic vector space setup?
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Introduction to Information Retrieval
Use angle instead of distance
 Rank documents according to angle with query
 Thought experiment: take a document d and append it to
itself. Call this document d′. d′ is twice as long as d.
 “Semantically” d and d′ have the same content.
 The angle between the two documents is 0, corresponding
to maximal similarity . . .
 . . . even though the Euclidean distance between the two
documents can be quite large.
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Introduction to Information Retrieval
From angles to cosines
 The following two notions are equivalent.
 Rank documents according to the angle between query and
document in decreasing order
 Rank documents according to cosine(query,document) in
increasing order
 Cosine is a monotonically decreasing function of the angle
for the interval [0◦, 180◦]
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Introduction to Information Retrieval
Cosine
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Introduction to Information Retrieval
Length normalization
 How do we compute the cosine?
 A vector can be (length-) normalized by dividing each of its
components by its length – here we use the L2 norm:
 This maps vectors onto the unit sphere . . .
 . . . since after normalization:
 As a result, longer documents and shorter documents have
weights of the same order of magnitude.
 Effect on the two documents d and d′ (d appended to itself)
from earlier slide: they have identical vectors after lengthnormalization.
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Introduction to Information Retrieval
Cosine similarity between query and
document




qi is the tf-idf weight of term i in the query.
di is the tf-idf weight of term i in the document.
| | and | | are the lengths of and
This is the cosine similarity of and . . . . . . or,
equivalently, the cosine of the angle between and
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Introduction to Information Retrieval
Cosine for normalized vectors
 For normalized vectors, the cosine is equivalent to the dot
product or scalar product.
 (if
and
are length-normalized).
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Introduction to Information Retrieval
Cosine similarity illustrated
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Introduction to Information Retrieval
Cosine: Example
term frequencies (counts)
How similar are
these novels? SaS:
Sense and
Sensibility PaP:
Pride and
Prejudice WH:
Wuthering
Heights
term
AFFECTION
JEALOUS
GOSSIP
WUTHERING
SaS
115
10
2
0
PaP
58
7
0
0
WH
20
11
6
38
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Introduction to Information Retrieval
Cosine: Example
term frequencies (counts)
term
SaS PaP WH
AFFECTION
115 58 20
JEALOUS
10
7 11
GOSSIP
2
0
6
WUTHERING
0
0 38
log frequency weighting
term
AFFECTION
JEALOUS
GOSSIP
WUTHERING
SaS PaP
3.06 2.76
2.0 1.85
1.30
0
0
0
WH
2.30
2.04
1.78
2.58
(To simplify this example, we don't do idf weighting.)
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Introduction to Information Retrieval
Cosine: Example
log frequency weighting
SaS
PaP
WH
log frequency weighting &
cosine normalization
term
SaS PaP
WH
AFFECTION 3.06
JEALOUS
2.0
GOSSIP
1.30
WUTHERING
0
2.76
1.85
0
0
2.30
2.04
1.78
2.58
AFFECTION
JEALOUS
GOSSIP
WUTHERING
term
0.789
0.515
0.335
0.0
0.832
0.555
0.0
0.0
0.524
0.465
0.405
0.588
 cos(SaS,PaP) ≈
0.789 ∗ 0.832 + 0.515 ∗ 0.555 + 0.335 ∗ 0.0 + 0.0 ∗ 0.0 ≈ 0.94.
 cos(SaS,WH) ≈ 0.79
 cos(PaP,WH) ≈ 0.69
 Why do we have cos(SaS,PaP) > cos(SAS,WH)?
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Introduction to Information Retrieval
Computing the cosine score
58
Introduction to Information Retrieval
Components of tf-idf weighting
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Introduction to Information Retrieval
tf-idf example








We often use different weightings for queries and documents.
Notation: ddd.qqq
Example: lnc.ltn
document: logarithmic tf, no df weighting, cosine
normalization
query: logarithmic tf, idf, no normalization
Isn’t it bad to not idf-weight the document?
Example query: “best car insurance”
Example document: “car insurance auto insurance”
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Introduction to Information Retrieval
tf-idf example: Inc.Itn
Query: “best car insurance”. Document: “car insurance auto insurance”.
Key to columns: tf-raw: raw (unweighted) term frequency, tf-wght:
logarithmically weighted term frequency, df: document frequency, idf:
inverse document frequency, weight: the final weight of the term in the
query or document, n’lized: document weights after cosine normalization,
product: the product of final query weight and final document weight
1/1.92 ≈ 0.52
1.3/1.92 ≈ 0.68 Final similarity score between query and
document:  i wqi · wdi = 0 + 0 + 1.04 + 2.04 = 3.08 Questions?
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Introduction to Information Retrieval
Summary: Ranked retrieval in the vector space
model
 Represent the query as a weighted tf-idf vector
 Represent each document as a weighted tf-idf vector
 Compute the cosine similarity between the query vector and
each document vector
 Rank documents with respect to the query
 Return the top K (e.g., K = 10) to the user
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Introduction to Information Retrieval
Take-away today
 Ranking search results: why it is important (as opposed to just
presenting a set of unordered Boolean results)
 Term frequency: This is a key ingredient for ranking.
 Tf-idf ranking: best known traditional ranking scheme
 Vector space model: One of the most important formal
models for information retrieval (along with Boolean and
probabilistic models)
63
Introduction to Information Retrieval
Resources
 Chapters 6 and 7 of IIR
 Resources at http://ifnlp.org/ir
 Vector space for dummies
 Exploring the similarity space (Moffat and Zobel, 2005)
 Okapi BM25 (a state-of-the-art weighting method, 11.4.3 of IIR)
64

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